The pope's meeting with his political nemesis, commonly known as Cristina Kirchner, was over a dispute with Britain over the Falkland Islands. Argentina and Britain fought a 1982 war over the islands. Earlier this month, the islanders voted overwhelmingly to remain a British Overseas Territory.
Kirchner said she had asked for Francis' intercession to "facilitate dialogue" with Britain over the islands, but there was no immediate comment from the Vatican as to whether the pope would accept her request.
The pope has been quoted in the past as saying that Britain "usurped" the remote islands, which Argentina claims and calls the Malvinas.
Francis and Kirchner are longtime rivals. Francis had accused the president's populist government of demagoguery, while she called his position on gay adoptions reminiscent of the Middle Ages and the Inquisition.
But on Monday, the two seemed to have patched up their relationship. Kirchner gave the new pope a mate gourd and straw to hold the traditional Argentine tea that Francis loves, and he gave her a kiss.
"Never in my life has a pope kissed me!" Kirchner said afterward.
The Vatican is gearing up for Francis' installation Mass Tuesday, which was expected to be a simplified version of the 2005 installation Mass that brought Pope Benedict XVI to the papacy.
The Vatican has released details of Francis' coat of arms and official ring, both of which are in keeping with his simple style and harking back to popes past: The coat of arms is the same Jesuit-inspired one he used as archbishop of Buenos Aires, while the ring was once offered to Pope Paul VI, who presided over the second half of the Second Vatican Council, the church meetings that modernized the church.
Francis will officially receive the Fisherman's Ring, marking the first time it will be worn by a non-European in 1,300 years.
The Associated Press contributed to this report.